It's certainly the Council's first major mistake. If we're going by the Jedi Order as a whole, then Mistake # 1 goes to Qui-Gon.
Someone (I think it was Stravo) posed an interesting question a few years back: Was Anakin always arrogant or did Qui-Gon plant the seeds?
I think that Anakin probably would've done the best under Qui-Gon, because Qui-Gon was the only example of a Jedi who seemed to trust him and believe in him, without blindly following Yoda's dogma. The way the series is structured seems to be on Qui-Gon's side. The Jedi Council is introduced in TPM as a stagnant and out of touch, while Qui-Gon was the kind and independent maverick. The Jedi screw up and are outmaneuvered by Palpatine in AOTC and ROTS. Obi-Wan admits that he failed Anakin during their duel on Mustafar, and at the end both he and Yoda receive training from the ghost of Qui-Gon Jinn.
Obi-Wan seems to have lightened up a lot by ROTS, forming a more trusting relationship with Anakin. But in AOTC, he's strict, kind of a jerk, and even a hypocrite (because he was a good disciple of the Council's ways). People always point out Anakin being angsty and whiny in AOTC, but during the chase on Coruscant he's
the go-getter who keeps a positive attitude while getting things done. Obi-Wan criticizes Anakin for his recklessness, even though Obi-Wan's the one who thought it was a good idea to jump out a skyscraper window (a decision that nearly gets him killed, until Anakin saves him). He admonishes Anakin for losing his lightsaber during the fight as if that could've been helped, but later loses his own lightsaber while fighting Jango Fett. Obi-Wan in AOTC acts just like a lot of stubborn, critical, and close-minded older men that I've seen in real life. One thing that I really liked about the Obi-Wan/Anakin relationship was that they were both
wrong, and both right, in different ways. Their character flaws fed on each other. Obi-Wan's strict and demeaning mentoring style embittered Anakin, while Anakin lacked the ability to emotionally get over it.
"They're not triangular, but they are more or less blade-shaped"
- Thrawn McEwok on the shape of Bakura
"Lovely. It's known as impugning character regarding statement of professional qualifications' in the legal world"
- Karen Traviss, crying libel because I said that no soldier she interviewed would claim that he can take on billion-to-one odds
"I've already laid out rules for this thread that we're not going to make these evidential demands"
- Dark Moose on supporting your claims